Download Speed of an Electromagnetic wave using Light Amplification by

May 2013
1
Measurement of Speed of Light
Shawn Kann
Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132
Abstract: Previous studies have shown that speed of light, c can be determined by a time-of-flight method. In this study, we measured the
speed of light by using a laser driven by a function generator. The distance that light traveled was varied, and the change in the time delay in light
generated by the function generator was compared with light emitted by the laser after it has traveled over distance. The experimentally
measured value of c = 2.4x108 m/s is consistent in orders of magnitude with the current national institute of science and technology (NIST) value.
INTRODUCTION/THEORY
The speed of light, c is an important constant used
in many physics formulas and can be derived from
Maxwell’s equations. In regions of space where
there is no charge or current, Maxwell’s equations
read [1],



B
(i)   E  0, (iii)   E  
,
t


E
(ii)   B  0, (iv)   B   0  0
.
t
Preceding equations constitute a set of coupled,
first-order, partial differential equations for E and
B. They can be decoupled by applying the curl to
(iii) and (iv) [2],




 B 
2
  (  E )  (  E )   E     
 t 



2E
  (  B)    0 0 2 ,
t
t





E 
2

  (  B)  (  B)   B      0 0


t





2B
  0 0 (  E )    0 0 2 .
t
t


Or, since   E  0 and   B  0,


2
2



E

B
 2 E   0 0 2 ,  2 B   0 0 2 .
t
t
We now have second order decoupled differential
equations for E and B [2]. In vacuum, then, each
Cartesian component of E and B satisfies the threedimensional wave equation,
2 f 
1 2 f
.
v 2 t 2
So, Maxwell’s equations imply that empty space
supports the propagation of electromagnetic waves,
traveling at a speed
v
1
 0 0
 3.00  10 8 m / s,
which happens to be precisely the velocity of light, c
[2].
Maxwell’s theory predicts the extremely
important idea that electrical disturbances are
transverse waves of electric and magnetic fields.
The velocity of propagation depends upon vacuum
permittivity  0 and permeability µ0, which can be
determined purely by electrical measurements and
should be equal to the velocity of light.
May 2013
2
this experiment.
The reflected output was
amplified by sending the output signal thru
TDS1002 Tektronix Digital Oscilloscope to determine
the time delay in the received light signal. The
optical components included a 10 cm lens to focus
the laser beam and a front-surface mirror to reflect
the beam. To generate a sine wave, we used 33120A
Agilent Function Generator. In addition, a leveling
base to hold laser, lens, and detector were also used.
The optical setup of this experiment is shown in
figure 2.
Figure 1. Photograph of a typical display on the dual
trace oscilloscope showing the sharp shift between
the signal generator and the signal of the reflected
beam.
EXPERIMENTAL METHODS
In this experiment, we used “time-of-flight”
method to measure the speed of light. The original
attempt to do this was by Galileo, who used flags
and lights, with human operators doing the timing
measurements [3]. In this experiment, we used a
special laser whose intensity was modulated with
the help of a signal generator. Using the
oscilloscope we then measured the change in the
time delay in light generated by the signal generator
and compared it with the light emitted by the laser
after it traveled some distance (figure 1.). The
peaks of the signals in figure 1 are slightly shifted.
by t . By repeating measurements for long
distances and short distances, we arrived at a time
t to travel a distance x ,
2d x

 t  10ns.
c
c
Plotting x vs. t and performing fit to a straight
line allowed us to compute the speed of light
constant as 1/c.
A ML868 Metrologic Laser with intensity
modulation capability was used as the source of
light. Metrologic detector unit with photodiodes and
amplifier was chosen to detect the reflected light for
Figure 2: Optical setup used to determine c.
The signal generator was configured to provide a
sine wave at 800 kHz with peak-to-peak amplitude
of 900mV, to modulate the laser intensity
RESULTS
By plotting time delay versus distance and using a
least-square-fit program in MATLAB, we can
determine a close fitting slope as,
c
1
m
(1)
The graph in (figure 3.) has a slope with a value of
about 0.10745. Plugging into equation 1 yields,
c
1
1
inch
m

 2.40  108
m 0.10745 ns
s
May 2013
3
2.9979x108 m/s. Comparing the discrepancy and
the accepted value the descrpency is,
2.9979  108 m / s  2.3639  108 m / s
 100  27%
2.3639  108 m / s
The large descrepancy could be attributed to
sources of error as discussed shortly in conclusion.
CONCLUSION
Figure 3: Plot of timedelay vs. distance.
ANALYSIS
The experimental value of speed of light constant
is obtained using equation (1) where the value of
the slope yields the experimental value of the speed
of light constant,
c
1
1
inch
m

 2.40  108
m 0.10745 ns
s
which agree in order of magnitude with the
currently accepted value of 299,792,458 m/s as
given by NIST.
In addition, the graph in figure shows Δm, the
uncertainy in the measured slope,
m  0.0098
ns
inch
m 0.0098

 9%
m 0.1075
c m

 9%
c
m
c  0.2  108 m / s
c  (2.4  0.2)  108 m / s
The theoratical value of the speed of light constant
according to NIST is 2.9979x108 m/s and the
experimental value of the speed of light constant is
This study has demonstrated that the time of
flight method is a suitable method to determine
speed of light, c. While the experimental result of
2.40x108 m/s is in good agreement with the
accepted value of 2.9979x108 ms-1 in orders of
magnitude, there was a relative discrepancy of 27%.
This discrepancy can be attributed to the possible
systematic source of error inside the oscilloscope.
There is also random error associated with our
subjective judgment of tmeasure. Moreover, when the
distance between detector and reflected mirror is
large, it is hard to focus the reflected laser beam
into the signal detector, in order to get high enough
signal to noise ratio.
Acknowledgement
The author thanks Dr. Weining Man for providing
the Matlab Plot1 script. The author also thanks lab
partner, Leung Mike. Figures one and two are
courtesy of Department of Physics at San Francisco
State University, CA.
References
[1] James Clerk Maxwell, “A Dynamical Theory of
the Electromagnetic Field,” Philosophical
Transactions of the Royal Society of London , 155 ,
459-512 (1865)
[2] David J. Griffiths, Introduction to
Electrodynamics (3rd ed, Addison Welsley, 1999),
Chap. 9, p. 375.
May 2013
[3] Maurice A. Finocchiaro, Retrying Galileo
(Universiy of California Press, 2007), Chap. 3, p.63.
4
State University, CA.